An optimization of composite sandwich structure with passive vibration absorber for vibration suppression

Abstract

This article proposes the use of a support as a passive vibration absorber to a composite sandwich structure for vibration suppression of satellite structures. Based on continuous mass distributions, an approximate formulation is presented for conducting vibration (modal, frequency response) analyses of the composite sandwich structure with the support. This formulation is derived by the Ritz method; verified for accuracy and computational efficiency by comparing finite element analyses. Finally, we perform optimization of the composite sandwich structure with passive vibration absorber by the present method. This optimization is conducted to applying satellite structures for maximizing vibration suppression performance in limited mass. The optimization result allows a database to be obtained on the vibration characteristics of composite sandwich structures with passive vibration absorber for applying aerospace applications. Consequently, it is concluded that the approximate formulation is well suited to vibration analyses of composite sandwich structures with passive vibration absorber due to their relative simplicity and computational efficiency.

Keywords

Composite sandwich structure passive vibration absorber approximate formulation Ritz method modal analysis frequency response analysis optimization

Introduction

In the aerospace industry, the latest satellites are required to quickly move and rotate to meet the demands of getting high quality images over wide areas. However, low stiffness structures often vibrate when satellites move quickly [1]. Additionally, these vibrations are transmitted to satellite structures, which can result in a delay in the stabilization time and the deterioration of the optical imaging collected by the satellite [2]. For these reasons, satellite structures should be designed with vibration suppression in mind by using high stiffness structures with vibration absorbers. Accordingly, composite sandwich structures have been used increasingly more for satellite structures because of several advantages, such as their high stiffness to weight ratio and high structural damping [3]. When applied to satellite structures, the structural/vibration characteristics of the composite sandwich structures should be predicted by analytical/numerical methods. Traditional studies have dealt with structural/vibration characteristics using analytical/numerical/experimental methods. Peter S et al. [4] investigated the structural characteristics of sandwich structures using finite element analysis. Kumar SS et al. [5] addressed the structural performances of carbon composite sandwich material with agglomerated cork core using finite element analysis. Wang MN et al. [6] focused on the in-plane shear response and failure mode of large size honeycomb sandwich composites that consisted of plain weave carbon fabric laminate skins and aramid paper core using numerical/experimental methods. Venkatesan KR et al. [7] investigated the thermomechanical performance of complex composite sandwich structures with adhesively bonded metallic fittings when exposed to extreme temperature ranges using the finite element method. Maha MAL and A Okasha EN [8] addressed the vibration characteristics (natural frequency/mode) of sandwich beams with various boundary conditions using the finite element approach. Hakim B et al. [9] presented an accurate shell finite element formulation to model composite shell structures with embedded viscoelastic and piezoelectric layers and an integrated active damping control mechanism to calculate vibration characteristics. Cho HK and Rhee J [10] investigated the complex vibration characteristics of an actual spacecraft structure using finite element analysis in conjunction with experimental data. The finite element method and experimental method have advantages when investigating the detailed structural/vibration characteristics of composite sandwich structures. However, these methods are not suitable for parametric/sensitivity analyses or optimization, both of which need a number of iterative processes owing to the difficulty of modifying a model and the amount of time required to reach a solution. Therefore, a number of recent studies on the structural/vibration characteristics of composite sandwich structures have been conducted using the homogenization method and other theoretical approaches owing to their relative simplicity and computational efficiency. The mechanical characteristics of honeycomb sandwich structures were investigated using simplified theories and models and the finite element method [11 –14]. Hassan I [15] presented a simplified methodology for the analysis of sandwich structures using the homogenization method and investigated the equivalent stiffness of sandwich structures with complex hexagonal cores using numerical/experimental methods. Yong-jing W et al. [16] investigated the vibration characteristics of sandwich panels with a hierarchical composite honeycomb sandwich core using the finite element method by proposing an equivalent model (two-dimensional model). Lopatin AV and Morozov EV [17] used the continuous method to address the vibration characteristics of composite sandwich plates. Serdoun SMN and SM Hamza C [18] presented theoretical research on the free vibration analysis of composite thick rectangular plates using Reddy’s higher order shear deformation theory (HSDT). Shanshan S et al. [19] proposed combining the complementary properties of honeycomb cores and grid cores in a composite sandwich panel with a honeycomb–grid hybrid core to enhance the structural performance of composite sandwich panels. Guo-dong X et al. [20] presented a theoretical approach for the free vibration of a composite sandwich beam with a graded corrugated lattice core based on a continuous homogeneous theory. Manish KD and Satyajit P [21] theoretically presented the shear-based attenuation of the vibration of an annular sandwich plate using two different shear mode piezoelectric fiber composite actuators, namely, the Shear Actuated Fiber Composite and the Balanced Laminate of piezoelectric fiber composite. Chyanbin H [22] investigated smart composite sandwich beams with surface bonded piezoelectric sensors and actuators by defining the theoretical formulation for the vibration suppression. However, these studies did not optimize a composite sandwich structure with a passive vibration absorber or the torsional spring boundary condition based on the theoretical formulation. In addition, it is important to quickly predict vibration characteristics because the predicted vibration performances should be the guidelines of the conceptual and preliminary designs of satellite structures. Although active vibration control absorbers are efficient for vibration suppression, a support as a passive vibration absorber is suitable because of the priority of the reliability of satellite structures [1]. Thus, we should define a theoretical model based on the continuous mass distributions to perform the vibration analysis and optimize the simplicity and computational efficiency of designs.

This article proposes the use of a support as a passive vibration absorber to a composite sandwich structure for the vibration suppression of satellite structures. Based on continuous mass distributions, an approximate formulation is presented for conducting the vibration analyses of the composite sandwich structure with a passive vibration absorber. This formulation is derived by defining the displacement functions for satisfying the boundary conditions. The Ritz method is applied to solve the nondimensional governing equations. The validity of the present method is verified by comparing the results of the finite element analysis (ABAQUS). Finally, we optimize a composite sandwich structure with a passive vibration absorber using the present method. The optimization result allows a database of the vibrational characteristics of composite sandwich structures with passive vibration absorbers that can be applied in aerospace applications to be obtained. Consequently, it is concluded that the approximate formulation is well suited for the vibration analyses of composite sandwich structures with passive vibration absorbers due to their relative simplicity and computational efficiency.

Problem statement and analysis

This article deals with the vibration analysis of a composite sandwich structure with a passive vibration absorber that can be applied to satellite structures. In the application to satellite structures, there are two types (deployed/undeployed modes) of composite sandwich structures, as shown in Figure 1. Accordingly, a schematic of a vibration problem is first defined, as shown in Figure 2. The deployed composite sandwich structure is subjected to an excitation force at the end point, with torsional spring boundary conditions at two points, and it is also equipped with a passive vibration absorber, as shown in Figure 3. The passive vibration absorber has a tubular cross section. The undeployed composite sandwich structure has a torsional spring at two points and has simply supported boundary conditions at two points. L_a and L_b are the length/width of the composite sandwich structure, respectively; and L_P and θ_P are the position/angle of the passive vibration absorber, respectively.

Fundamental equations

In their application to satellite structures, the composite sandwich structure consists of CFRP layers, a honeycomb core layer, a solar cell layer, and a coverglass layer, as shown in Figure 4. The Kirchhoff-Love assumption [23] applicable to an elastic plate is adopted to define the stiffness of the composite sandwich structure. Additionally, this structure is replaced with a continuous orthotropic plate with equivalent stiffness parameters. The membrane/membrane-bending/bending effective stiffness matrices can be derived as follows:

\begin{array}{l} A_{i j} = \sum_{k = 1}^{n} {[\bar{Q}]}_{k} (z_{k} - z_{k - 1}), B_{i j} = \sum_{k = 1}^{n} {[\bar{Q}]}_{k} (\frac{z_{k}^{2} - z_{k - 1}^{2}}{2}), \\ D_{i j} = \sum_{k = 1}^{n} {[\bar{Q}]}_{k} (\frac{z_{k}^{3} - z_{k - 1}^{3}}{3}) \end{array}

(1)

{[\bar{Q}]}_{k} = {[T]}_{k}^{- 1} {[Q]}_{k} {({[T]}_{k}^{T})}^{- 1}

(2)

{[Q]}_{k} = {[\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}]}_{k}

(3)

{[T]}_{k} = [\begin{matrix} \cos^{2} θ_{k} & \sin^{2} θ_{k} & 2 \cos θ_{k} \sin θ_{k} \\ \sin^{2} θ_{k} & \cos^{2} θ_{k} & - 2 \cos θ_{k} \sin θ_{k} \\ - \cos θ_{k} \sin θ_{k} & \cos θ_{k} \sin θ_{k} & \cos^{2} θ_{k} - \sin^{2} θ_{k} \end{matrix}]

(4)

where θ_k is the orientation of the fibers. Q_ij is the reduced stiffness defined in terms of the material properties of the layers as follows:

Q_{11} = \frac{E_{1}}{1 - v_{12} v_{21}}, Q_{22} = \frac{E_{2}}{1 - v_{12} v_{21}}, Q_{12} = \frac{v_{12} E_{2}}{1 - v_{12} v_{21}}, Q_{66} = G_{12}

(5)

where E₁ and E₂ mean the modulus of the elasticity in directions 1 and 2, respectively; G₁₂ means the shear modulus; and v_ij means the Poisson’s ratio.

In the vibration analysis, the membrane-bending effective stiffness matrix should be involved because of the dummy components for the structural stiffness (Figure 3). However, component B₁₂ of the membrane-bending effective stiffness matrix is assumed to be zero because it is relatively small compared to other components. The honeycomb core layer consists of a regular hexagonal honeycomb with a uniform thickness, as shown in Figure 5. According to the traditional method [19], the equivalent elastic parameters for the honeycomb core can be calculated as follows:

E_{1}^{*} = {(\frac{t_{h}}{l_{h}})}^{3} \frac{\cos θ_{h}}{(\frac{h_{h}}{l_{h}} + \sin θ_{h}) \sin^{2} θ_{h}} E_{h}

(6)

E_{2}^{*} = {(\frac{t_{h}}{l_{h}})}^{3} \frac{(\frac{h_{h}}{l_{h}} + \sin θ_{h})}{\cos^{3} θ_{h}} E_{h}

(7)

G_{12}^{*} = {(\frac{t_{h}}{l_{h}})}^{3} \frac{(\frac{h_{h}}{l_{h}} + \sin θ_{h})}{{(\frac{h_{h}}{l_{h}})}^{2} (1 + 2 \frac{h_{h}}{l_{h}}) \cos θ_{h}} E_{h}

(8)

ρ^{*} = \frac{(l_{h} + h_{h}) t_{h}}{(l_{h} \sin θ_{h} + h_{h}) l_{h} \cos θ_{h}} ρ_{h}

(9)

where E_h and ρ_h are the modulus of elasticity and the density of the honeycomb material, respectively, and h_h and l_h vertical/angled length of a honeycomb core, and t_h and θ_h are thickness of honeycomb walls and inclined angle of a honeycomb core.

Displacement functions of the composite sandwich structure with a passive vibration absorber

The in-plane/bending displacement functions of the composite sandwich structure are required for solving the governing equation using the Ritz method. To describe the mechanical behavior of this structure with a torsional spring and simply supported boundary conditions, the in-plane/bending displacement functions should be the comparison functions satisfying the geometrical/natural boundary conditions below.

At two points, the torsional spring boundary conditions (deployed composite sandwich structure) are

u (0, 0) = u (0, L_{b}) = v (0, 0) = v (0, L_{b}) = w (0, 0) = w (0, L_{b}) = 0

(10)

k_{T} \frac{\partial w (0, 0)}{\partial x} = M_{x} (0, 0)

(11)

k_{T} \frac{\partial w (0, L_{b})}{\partial x} = M_{x} (0, L_{b})

(12)

N_{x} (L_{a}, y) = M_{x} (L_{a}, y) = S_{x} (L_{a}, y) = 0

(13)

N_{y} (x, 0) = N_{y} (x, L_{b}) = M_{y} (x, 0) = M_{y} (x, L_{b}) = S_{y} (x, 0) = S_{y} (x, L_{b}) = 0

(14)

k_{P} δ = S_{x} (L_{p}, \frac{L_{b}}{2})

(15)

At four points, the torsional spring and simply supported boundary conditions (undeployed composite sandwich structure) are

u (0, 0) = u (0, L_{b}) = v (0, 0) = v (0, L_{b}) = w (0, 0) = w (0, L_{b}) = 0

(16)

u (L_{a}, 0) = u (L_{a}, L_{b}) = v (L_{a}, 0) = v (L_{a}, L_{b}) = w (L_{a}, 0) = w (L_{a}, L_{b}) = 0

(17)

k_{T} \frac{\partial w (0, 0)}{\partial x} = M_{x} (0, 0)

(18)

k_{T} \frac{\partial w (0, L_{b})}{\partial x} = M_{x} (0, L_{b})

(19)

M_{x} (L_{a}, 0) = M_{x} (L_{a}, L_{b}) = 0

(20)

N_{y} (x, 0) = N_{y} (x, L_{b}) = M_{y} (x, 0) = M_{y} (x, L_{b}) = S_{y} (x, 0) = S_{y} (x, L_{b}) = 0

(21)

where N, M and S are in-plane force/moment resultants and a shear force, and δ, k_P, and k_T are the displacement/stiffness of the passive vibration absorber and the stiffness of the torsional spring, respectively. These parameters can be derived as follows:

δ = \frac{w (L_{p}, \frac{L_{b}}{2})}{\cos θ_{p}}

(22)

k_{P} = \frac{\cos θ_{p}}{L_{P}} π \sum_{k = 1}^{n} {[\bar{Q}]}_{k} (\frac{r_{k}^{2} - r_{k - 1}^{2}}{2})

(23)

The mechanical behavior of the composite sandwich structure is decomposed into components u, v, and w parallel to the x, y, and z coordinates, respectively (Figure 2). The in-plane force/moment resultants N and M can be generally written as follows:

N_{x} = A_{11} \frac{\partial u}{\partial x} + A_{12} \frac{\partial v}{\partial y} + B_{11} \frac{\partial^{2} w}{\partial x^{2}}

(24)

N_{y} = A_{12} \frac{\partial u}{\partial x} + A_{22} \frac{\partial v}{\partial y} + B_{22} \frac{\partial^{2} w}{\partial y^{2}}

(25)

M_{x} = B_{11} \frac{\partial u}{\partial x} + D_{11} \frac{\partial^{2} w}{\partial x^{2}} + D_{12} \frac{\partial^{2} w}{\partial y^{2}}

(26)

M_{y} = B_{22} \frac{\partial v}{\partial y} + D_{12} \frac{\partial^{2} w}{\partial x^{2}} + D_{22} \frac{\partial^{2} w}{\partial y^{2}}

(27)

To apply the decomposition method to geometry/natural boundary conditions, the in-plane/bending displacement functions can be defined as follows:

U (x, y, t) = u (x, y) e^{j ω t}

(28)

V (x, y, t) = v (x, y) e^{j ω t}

(29)

W (x, y, t) = w (x, y) e^{j ω t}

(30)

u (x, y) = \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} [\sum_{i = 1}^{m} a_{i} F_{i} (x) + \sum_{j = 1}^{n} a_{j + m} F_{j} (x) (\frac{y - L_{b} / 2}{L_{b}})]

(31)

\begin{array}{l} v (x, y) = - \frac{D_{12}}{B_{22}} {\sum_{i = 1}^{m} a_{i} \frac{\partial^{2} f_{i} (x)}{\partial x^{2}} L_{b} [2 {(\frac{y}{L_{b}})}^{3} - 3 {(\frac{y}{L_{b}})}^{2} + \frac{y}{L_{b}}] \\ + \sum_{j = 1}^{n} a_{j + m} \frac{\partial^{2} f_{j} (x)}{\partial x^{2}} (\frac{y^{2} - L_{b} y}{2 L_{b}})} \end{array}

(32)

w (x, y) = \sum_{i = 1}^{m} a_{i} f_{i} (x) + \sum_{j = 1}^{n} a_{j + m} f_{j} (x) (\frac{y - L_{b} / 2}{L_{b}})

(33)

where a_i is a displacement coefficient that can be determined, and m and n are the orders of the displacement functions.

Based on equations (10) to (21) and (24) to (27), the geometrical/natural boundary conditions can be rewritten as equations (34) to (36) and (39) to (43), respectively. To satisfy these boundary conditions, the in-plane/bending displacement functions are defined as polynomial functions, as follows.

1) At two points, the torsional spring boundary conditions (deployed composite sandwich structure) are as follows.

1.1) Geometrical boundary conditions

f_{i} (0) = F_{i} (0) = 0

(34)

1.2) Natural boundary conditions

k_{T} \frac{\partial f_{i} (0)}{\partial x} = B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} \frac{\partial F_{i} (0)}{\partial x} + D_{11} \frac{\partial^{2} f_{i} (0)}{\partial x^{2}}

(35)

\frac{\partial^{2} f_{i} (L_{a})}{\partial x^{2}} = \frac{\partial^{3} f_{i} (L_{a})}{\partial x^{3}} = 0

(36)

1.3) Spatial functions for the displacement functions

\begin{matrix} f_{i} (x) = {(\frac{x}{L_{a}})}^{i + 3} - \frac{(i + 3) (i + 2) (i + 1)}{6} {(\frac{x}{L_{a}})}^{3} + \frac{i (i + 3) (i + 2)}{2} {(\frac{x}{L_{a}})}^{2} \\ + (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11}) \frac{i (i + 3) (i + 2)}{L_{a} k_{T}} (\frac{x}{L_{a}}) \end{matrix}

(37)

F_{i} (x) = \frac{\partial f_{i} (x)}{\partial x} - (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11}) \frac{i (i + 3) (i + 2)}{L_{a}^{2} k_{T}}

(38)

2) At four points, the torsional spring and simply supported boundary conditions (undeployed composite sandwich structure) are defined as follows.

2.1) Geometrical boundary conditions

f_{i} (0) = F_{i} (0) = f_{i} (L_{a}) = F_{i} (L_{a}) = 0

(39)

2.2) Natural boundary conditions

k_{T} \frac{\partial f_{i} (0)}{\partial x} = B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} \frac{\partial F_{i} (0)}{\partial x} + D_{11} \frac{\partial^{2} f_{i} (0)}{\partial x^{2}}

(40)

\frac{\partial^{2} f_{i} (L_{a})}{\partial x^{2}} = 0

(41)

2.3) Spatial functions for the displacement functions

\begin{array}{l} \begin{array}{l} f_{i} (x) = {(\frac{x}{L_{a}})}^{i + 4} - \frac{1}{12 (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11}) + L_{a} k_{T}} \\ \times {[\frac{(i + 2) (i + 1)}{2} L_{a} k_{T} \end{array} \\ \begin{array}{l} + 3 (i + 1) (i + 4) (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11})] {(\frac{x}{L_{a}})}^{4} \\ - [i (i + 2) L_{a} k_{T} \end{array} \\ \begin{array}{l} + 4 i (i + 4) (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11})] {(\frac{x}{L_{a}})}^{3} \\ + \frac{i (i + 1)}{2} L_{a} k_{T} {(\frac{x}{L_{a}})}^{2} \end{array} \\ + i (i + 1) (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11}) (\frac{x}{L_{a}})} \end{array}

(42)

F_{i} (x) = \frac{\partial f_{i} (x)}{\partial x} + \frac{i (i + 1) (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11})}{L_{a} [12 (B_{11} \frac{A_{22}}{A_{12}} \frac{D_{12}}{B_{22}} + D_{11}) + L_{a} k_{T}]}

(43)

Theoretical modeling and analysis (modal analysis)

In this article, the nondimensional governing equations are defined by the Ritz method. First, the kinetic/potential energies and external work in terms of the displacement functions are derived by substituting the defined in-plane/bending displacement functions into mathematical formulations. The kinetic/potential energies and external work can be written as follows:

K E = \frac{1}{2} h_{e q} ρ_{e q} \iint ({\dot{U}}^{2} + {\dot{V}}^{2} + {\dot{W}}^{2}) d A + \frac{1}{2} m_{P} {\dot{δ}}^{2}

(44)

P E = U_{M} + U_{M B} + U_{B} + \frac{1}{2} k_{T} [{(\frac{\partial w (0, 0)}{\partial x})}^{2} + {(\frac{\partial w (0, L_{b})}{\partial x})}^{2}] + \frac{1}{2} k_{P} δ^{2}

(45)

U_{M} = \frac{1}{2} \iint [A_{11} {(\frac{\partial U}{\partial x})}^{2} + A_{22} {(\frac{\partial V}{\partial y})}^{2} + A_{66} {(\frac{\partial U}{\partial y} + \frac{\partial V}{\partial x})}^{2} + 2 A_{12} \frac{\partial U}{\partial x} \frac{\partial V}{\partial y}] d A

(46)

U_{M B} = \frac{1}{2} \iint [B_{11} \frac{\partial U}{\partial x} \frac{\partial^{2} W}{\partial x^{2}} + B_{22} \frac{\partial V}{\partial y} \frac{\partial^{2} W}{\partial y^{2}} + 2 B_{66} (\frac{\partial U}{\partial y} + \frac{\partial V}{\partial x}) \frac{\partial^{2} W}{\partial x \partial y}] d A

(47)

U_{B} = \frac{1}{2} \iint [D_{11} {(\frac{\partial^{2} W}{\partial x^{2}})}^{2} + D_{22} {(\frac{\partial^{2} W}{\partial y^{2}})}^{2} + 4 D_{66} {(\frac{\partial^{2} W}{\partial x \partial y})}^{2} + 2 D_{12} \frac{\partial^{2} W}{\partial x^{2}} \frac{\partial^{2} W}{\partial y^{2}}] d A

(48)

E W = w (L_{a}, \frac{L_{b}}{2}) f_{0} e^{j ω t}

(49)

where ρ_eq and h_eq are the equivalent density/height of the composite sandwich structure, respectively; m_P is the mass of the passive vibration absorber; and f₀ is the magnitude of the force.

The kinetic/potential energies and external work are minimized with respect to the displacement coefficients according to the principle of the minimum total energy [24]. Using the derivatives of equation (50), an equation can be derived in terms the displacement coefficients, which is represented by the following:

\frac{\partial}{\partial c} (P E^{*} - ω_{n}^{2} K E^{*}) = 0

(50)

K E^{*} = \frac{K E}{e^{- j ω t}}, P E^{*} = \frac{P E}{e^{- j ω t}}

(51)

This procedure gives the linear system (m + n) $\times$ (m + n) governing equation. Moreover, the orders (m, n) of the displacement functions are determined by convergence analysis.

[K] [a] = ω_{n}^{2} [M] [a]

(52)

[K] = [\begin{matrix} K_{11} & \dots & K_{1 m + n} \\ ⋮ & ⋱ & ⋮ \\ K_{m + n 1} & \dots & K_{m + n m + n} \end{matrix}]

(53)

[M] = [\begin{matrix} M_{11} & \dots & M_{1 m + n} \\ ⋮ & ⋱ & ⋮ \\ M_{m + n 1} & \dots & M_{m + n m + n} \end{matrix}]

(54)

[a] = {[\begin{matrix} a_{1} & \dots & a_{m} & a_{m + 1} & \dots & a_{m + n} \end{matrix}]}^{T}

(55)

where [M] and [K] are square matrices made up of the elements determined from the parameters of the composite sandwich structure with the passive vibration absorber. [a] is the column matrix of an eigenvector, which includes the displacement coefficients. The approximate natural frequencies and displacement coefficients will be obtained after the generalized eigenvalue problem, equation (52), is solved.

Theoretical modeling and analysis (frequency response analysis)

In this section, we deal with the dynamic modeling of the deployed composite sandwich structure with a passive vibration absorber for vibration suppression. Based on the defined displacement functions, the dynamic model is derived as follows:

(- ω^{2} [M] + j ω [C] + [K]) [a] e^{j ω t} = [F] e^{j ω t}

(56)

where [C] and [F] are the damping/force matrices of the deployed composite sandwich structure with the passive vibration absorber, respectively, which are defined by the following.

[C] = {[C]}_{CFRP} + {[C]}_{Honeycomb} + {[C]}_{Absorber}

(57)

{[C]}_{CFRP} = α_{CFRP} {[M]}_{CFRP} + β_{CFRP} {[K]}_{CFRP}

(58)

{[C]}_{Honeycomb} = α_{Honeycomb} {[M]}_{Honeycomb} + β_{Honeycomb} {[K]}_{Honeycomb}

(59)

{[C]}_{Absorber} = 2 ζ_{CFRP} \sqrt{\frac{k_{P}}{m_{P}}} δ^{2}

(60)

[F] = f_{0} [\begin{matrix} f_{1} (L_{a}) \\ ⋮ \\ f_{m} (L_{a}) \\ 0 \\ 0 \\ 0 \end{matrix}]

(61)

The damping matrices of the CFRP/honeycomb core layers are assumed to the linear combination of the mass/stiffness matrices of a Rayleigh damping model. ζ_CFRP means the damping ratio of the CFRP layers, and it is experimentally selected by the traditional method [25]. α and β respectively represent the mass/stiffness proportional damping coefficients, which are collectively known as the Rayleigh damping coefficients. The Rayleigh damping coefficients can be calculated by two orders of the reference frequency and damping ratios as follows:

[\begin{matrix} α \\ β \end{matrix}] = \frac{2 ω_{i} ω_{j}}{ω_{j}^{2} - ω_{i}^{2}} [\begin{matrix} ω_{j} & - ω_{i} \\ - \frac{1}{ω_{j}} & \frac{1}{ω_{i}} \end{matrix}] [\begin{matrix} ζ_{i} \\ ζ_{j} \end{matrix}]

(62)

To calculate the Rayleigh damping coefficients, first, the two orders of the reference frequency are selected. Second, the natural frequencies and the damping ratio of honeycomb core layers are experimentally selected using the traditional method [26]. Based on the dynamic model, the nondimensional displacement at the point subjected to the excitation force can be defined by the following equations:

a^{*} = | \frac{{[K]}^{*} {[a]}^{*}}{F_{0}} |

(63)

K_{i j}^{*} = {\begin{array}{l} K_{i j} - \frac{k_{P}}{\cos^{2} θ_{P}} f_{i} (L_{P}) f_{j} (L_{P}) i, j \leq m \\ 0 i > m or j > m \end{array}

(64)

{[a]}^{*} = {(- ω^{2} [M] + j ω [C] + [K])}^{- 1} [F]

(65)

In this article, this nondimensional displacement is selected as the basic guideline to optimize the vibration suppression of the deployed composite sandwich structure.

Verification of the present method via finite element analysis

To verify the validity of the theoretical modeling, the modal analyses of composite sandwich structures with passive vibration absorbers are performed using the present method and the finite element method.

Finite element model

The finite element models of composite sandwich structures with passive vibration absorbers are created by using HYPERMESH. This model and the preprocessor are illustrated in Figure 6. ABAQUS (Frequency, Linear perturbation) is used as the solver of the finite element analysis. The finite element model for the composite sandwich structure is described using shell elements (S4R) and the passive vibration absorber is described using truss elements (T3D2). The number of shell/truss elements is more than 70,000 to ensure the quality of the finite element analysis. The geometric parameters of the composite sandwich structure are presented in Table 1. The deployed composite sandwich structure is subjected to an excitation force at the end point with torsional spring boundary conditions at two points. The undeployed composite sandwich structure has torsional springs at two points and has simply supported boundary conditions at two points. The boundary conditions are described by multiple point constraints (MPCs) based on nodes. Additionally, the stiffness of the torsional spring is described by point elements using multiple point constraints. Furthermore, the material properties of the CFRP/honeycomb core/solar cell/coverglass layers are presented in Table 2.

Table 1.

Geometric parameters of the composite sandwich structures with the passive vibration absorber.

	Value		Unit
	Deployed mode	Un-deployed mode	Unit
Length	1,400		mm
Width	700
Thickness of a honeycomb layer	20
Vertical/angled length of a honeycomb core	10
Inclined angle of a honeycomb core	30		Degree
Thickness of honeycomb walls	0.5		mm
Ply thickness of a CFRP layer	0.2		mm
Laminated pattern of a CFRP layer	[0/90/90/0]		Degree
Thickness of a solar cell layer	1		mm
Thickness of a coverglass layer	1		mm
Stiffness of a torsional spring	10,000		N*mm/rad
Position of a passive vibration absorber	700	–	mm
Inner radius of a passive vibration absorber	5	–
Ply thickness of a passive vibration absorber	0.2	–
Laminated pattern of a passive vibration absorber	[0/90/90/0]	–	Degree

Table 2.

Material properties of CFRP/aluminum (honeycomb core).

		Value		Unit
		CFRP	Aluminum (honeycomb core)	Unit
Modulus of elasticity	E ₁	134	70	GPa
	E ₂	8.08	70	GPa
	G ₁₂	5.33	21	GPa
Poisson ratio	v ₁₂	0.33	0.3	–
Density	ρ	1,600	2,700	kg/m³

Modal analysis and discussion

In this section, we perform the modal analyses of the deployed/undeployed composite sandwich structures using the present method and the finite element method to verify the overall validity. Figures 7 and 8 present the natural frequencies and natural modes of the deployed/undeployed composite sandwich structures. Table 3 presents the results of the convergence analysis (ply thickness of a CFRP layer is 0.2 mm) according to the orders (m, n) of the displacement functions. These results confirm that the orders (m, n) should be more than 4 when calculating natural frequencies above the third mode. Accordingly, the orders (m, n) of the displacement functions are considered to be 4 by the convergence analysis. The first/second/third natural modes are the bending/torsional modes of shell. With the present method, the natural mode shapes of the deployed/undeployed composite sandwich structures are similar to those of the finite element method. Additionally, the results of the present method include the changes of the natural modes as the ply thickness increases, as shown in Figure 7. There is a difference of less than 7% between the natural frequencies from the present method and the finite element method. In addition, the solution time is 0.11 second for the modal analysis of the deployed composite structure using the present method and MATLAB. It is shorter than that of the finite element method, which takes 91 seconds, on the single CPU core used in this article.

Table 3.

Result of a convergence analysis.

Orders (m, n)	Value								Unit
	Deployed mode				Un-deployed mode
	1st mode	2nd mode	3rd mode	4th mode	1st mode	2nd mode	3rd mode	4th mode
1	19.65	84.16	–		50.63	63.39	–		Hz
2	18.04	26.30	72.04	148.8	30.55	46.26	226.8	240.8
3	18.02	25.58	71.88	122.2	29.50	45.36	129.4	147.8
4	17.93	25.55	71.67	122.1	29.05	45.00	116.6	135.4
5	17.93	25.45	71.46	120.0	28.88	44.88	115.4	134.4

Verification of the present method via experimental analysis

In this section, the validity of the theoretical modeling is additionally verified by experimental analyses. Experimental results are found in the literature survey of similar studies [27 –29]. In the experimental examples, the laminated composite plates commonly have a clamped boundary condition at an end edge. This boundary condition is formed by setting the stiffness of the torsional spring and the stiffness of the passive vibration absorber to infinity and zero, respectively. Additionally, the orders (m, n) of the displacement functions are considered to be 4 by the verification via finite element analysis. Figure 9 presents the natural frequencies of the laminated composite plates. With the present method, the natural mode shapes of the laminated composite plates are similar to those of the experimental method. There is a difference of less than 5% between the natural frequencies from the present method and the experimental method. Because the experimental examples are simpler, the experimental comparison results are better than those of the finite element analyses. Thus, this article verifies the relative simplicity and computational efficiency of the present method, and confirms that the vibration analyses with the displacement functions over an order of 3 are suitable for the structural design of a composite sandwich structure with a passive vibration absorber.

Optimization of the composite sandwich structure with a passive vibration absorber

In this section, an optimization problem for the vibration suppression of the deployed composite sandwich structure with a passive vibration absorber is formulated based on the stiffness requirements for satellite structure applications, and the optimization code is written using MATLAB. A schematic of the optimization problem is shown in Figure 10. First, the nondimensional displacement of the deployed composite sandwich structure is minimized by changing the position of the passive vibration absorber based on the initial parameters. Second, we perform the optimization to minimize the mass of the composite sandwich structure with the passive vibration absorber based on the determined position of the passive vibration absorber. A genetic algorithm (GA) is used as an optimization algorithm to design a composite sandwich structure with a passive vibration absorber. The genetic algorithm is suitable for finding the global optimum because the best design is always transferred from the previous generation to the next generation [30,31]. The generation and population sizes are set at 200 and 20, respectively.

Parametric analysis to minimize the nondimensional displacement

The parametric analysis is performed to minimize the nondimensional displacement at the point subjected to the excitation force by changing the position of the passive vibration absorber. In the parametric analysis, the deployed composite sandwich structure is subjected to an excitation force at the end point (L_a, L_b/2) with torsional spring boundary conditions at two points, and it is also equipped with a passive vibration absorber. The geometric parameters and range of the main parameter are presented in Table 4. Figure 11 presents the results of the parametric analysis. In these results, the nondimensional displacement nonlinearly decreases according to change in the position of the passive vibration absorber. Thus, it is confirmed that the nondimensional displacement is minimized at the lowest position, and so the position of the passive vibration absorber is determined to be 500 mm.

Table 4.

Geometric parameters for a parametric analysis.

	Value	Unit
Length	1,400	mm
Width	700
Thickness of a honeycomb layer	20
Vertical/angled length of a honeycomb core	2
Inclined angle of a honeycomb core	30	Degree
Thickness of honeycomb walls	0.066	mm
Ply thickness of a CFRP layer	0.2	mm
Laminated pattern of a CFRP layer	[0/90/90/0]	Degree
Thickness of solar cell/coverglass layers	1	mm
Stiffness of a torsional spring	10,000	N*mm/rad
Position of a passive vibration absorber	500 ∼ 900	mm
Inner radius of a passive vibration absorber	5
Ply thickness of a passive vibration absorber	0.2
Laminated pattern of a passive vibration absorber	[0/90/90/0]	Degree

Formulation of the optimization problem

The optimization problem of the Min Mass is defined as follows:

Minimize Mass (L_{a}, L_{b}, h_{e q}, ρ_{e q}, L_{P}, θ_{P}, r_{out}, r_{i n}) = L_{a} L_{b} h_{e q} ρ_{e q} + \frac{L_{P} (r_{out}^{2} - r_{i n}^{2}) π ρ_{CFRP}}{\cos θ_{P}}

(66)

\begin{array}{l} S .T . G_{1} = 1 - \frac{ω_{1}}{15} \leq 0 \\ G_{2} = 1 - \frac{ω_{2}}{15} \leq 0 \\ G_{3} = 1 - \frac{r_{out}}{r_{i n} + 0.8} \leq 0 \\ 10 \leq t_{h c} [m m] \leq 20 \\ 0.2 \leq t_{CFRP} [m m] \leq 1 \\ 5 \leq r_{i n} [m m] \leq 10 \\ 5 \leq r_{out} [m m] \leq 12 \\ 1, 000 \leq k_{T} [N \cdot m m / rad] \leq 20, 000 \end{array}

(67)

where ω₁ and ω₂ are the first modes of the deployed/undeployed composite sandwich structures, respectively. The objective function is the mass of the composite sandwich structure with the passive vibration absorber. The design parameters are the thicknesses of the CFRP/honeycomb core layers, the inner/outer radii of the passive vibration absorber, and the stiffness of the torsional spring. The area of the composite sandwich structure is fixed at 980,000 mm² (L_a = 1,400 mm and L_b = 700 mm), and it is calculated according to the required solar power. The design of the satellite structure requires sufficient stiffness such that the first mode of the composite sandwich structure is higher than 15 Hz [32].

Optimization results and discussion

Figures 12 to 16 present the objective function history and the design parameter histories. They show that the mass of the composite sandwich structure with the passive vibration absorber converges to 7.81 kg by increasing the generation and design parameters. Table 5 presents the optimized design parameters, the mass, the natural frequencies, and the maximum nondimensional displacement of the composite sandwich structure with the passive vibration absorber. Additionally, it is confirmed that these design parameters satisfy the stiffness requirements. Thus, we observe that the present method is very useful not only to optimize the design but also to carry out the vibration analysis of composite sandwich structures with passive vibration absorbers for satellite structures.

Table 5.

Optimized design variables, mass, non-dimensional displacement and natural frequencies/modes of the composite sandwich structure with the passive vibration absorber.

	Value	Unit
Mass	7.81	kg
Thickness of a honeycomb layer	15.8	mm
Ply thickness of a CFRP layer	0.2	mm
Stiffness of a torsional spring	12,724	N*mm/rad
Position of a passive vibration absorber	500	mm
Inner radius of a passive vibration absorber	5
Ply thickness of a passive vibration absorber	0.2
1st natural frequency (deployed mode)	16.3	Hz
1st natural mode (deployed mode)	Torsional mode	–
1st natural frequency (un-deployed mode)	27.3	Hz
1st natural mode (un-deployed mode)	Bending mode	–
Maximum non-dimensional displacement	115	dB

Conclusion

This article proposes the use of a support as a passive vibration absorber for a composite sandwich structure for vibration suppression in satellite structures. Based on continuous mass distributions, an approximate formulation is presented for conducting the vibration analyses of the composite sandwich structure with a passive vibration absorber. This method has advantages compared to existing numerical/experimental approaches, as described below.

Specific boundary conditions (with the torsional spring boundary condition at two points, and it is also equipped with the passive vibration absorber) frequently used in satellite structures are applied by defining the displacement functions assumed to be polynomial functions.

The nondimensional displacement can be investigated by involving the experimental damping coefficients of the CFRP/honeycomb core layers. This nondimensional displacement is selected as the basic guideline to optimize the vibration suppression of the composite sandwich structure.

There is high accuracy of more than 93% between the natural frequencies from the present method and the finite element method (ABAQUS) due to their relative simplicity and computational efficiency. With the present method, the natural mode shapes of the deployed/undeployed composite sandwich structures are similar to those of the finite element method.

There is a difference of less than 5% between the natural frequencies from the present method and the experimental method. Because the experimental examples are simpler, the experimental comparison results are better than those of the finite element analyses. The experimental results can be referred to in the literature survey of similar studies [27 –29].

This method has relative simplicity and computational efficiency compared to the finite element method and experimental method due to the advantages of easily modifying the model and the short solution time.

Finally, we optimize the composite sandwich structure with a passive vibration absorber using the present method. The nondimensional displacement of the deployed composite sandwich structure is minimized by changing the position of the passive vibration absorber based on the initial parameters. Next, we perform the optimization that minimizes the mass of the composite sandwich structure with the passive vibration absorber based on the determined position of the passive vibration absorber. These optimization results allow a database of the vibration characteristics of composite sandwich structures with passive vibration absorbers that can be applied in aerospace applications to be formed. Consequently, it is concluded that the approximate formulation is well suited to conduct the vibration analyses of composite sandwich structures with passive vibration absorbers due to their relative simplicity and computational efficiency.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yongha Kim

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